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If I play Time of Ice and immediately tap down creature A, then next turn tap down creature B, then play Rescue and replay Time of Ice to tap down creature C. On my opponent's turn after that, do creatures A and B stay tapped, since I still control Time of Ice, or do they untap since I stopped controlling that instance of time of ice after it went back to my hand briefly?

Below is the rules text for Time of Ice:

(As this Saga enters and after your draw step, add a lore counter. Sacrifice after III.)

I, II — Tap target creature an opponent controls. It doesn't untap
during its controller's untap step for as long as you control Time of
Ice.

1 Answer
1

The effects of Time of Ice’s first two chapter abilities expire if you lose control of it, even if you immediately regain control of it or cast another Time of Ice.

And the comprehensive rules that back up this ruling:

611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.

So the moment you return Time of Ice to your hand, that ability that was keeping A and B tapped has ended. That ability no longer has any effect on A or B; nothing that happens after that can change that.

Also, when you cast Time of Ice later, it is a completely different Time of Ice that has no connection to the one that you used on A and B.

400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence. There are nine exceptions to this rule:

(None of the exceptions apply here).

And when it says "for as long as you control Time of Ice", it means specifically that Time of Ice, and not any other objects with that name.

201.4. Text that refers to the object it’s on by name means just that particular object and not any other objects with that name, regardless of any name changes caused by game effects.